Q. 27 Evaluating triple integrals: Eac... [FREE SOLUTION]

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Chapter 13: Q. 27 (page 1082)

Evaluating triple integrals: Each of the triple integrals that follows represents the volume of a solid. Sketch the solid and evaluate the integral.
${鈭�}_{0}^{4} {鈭�}_{0}^{3 - \frac{3}{4} x} {鈭�}_{0}^{2 - \frac{1}{2} x - \frac{2}{3} y} d z d y d x$

Short Answer

Expert verified

${鈭�}_{0}^{4} {鈭�}_{0}^{3 - \frac{3}{4} x} {鈭�}_{0}^{2 - \frac{1}{2} x - \frac{2}{3} y} d z d y d x = 4$

Step by step solution

Given

We have to find the volume by using a triple integral.

${鈭�}_{0}^{4} {鈭�}_{0}^{3 - \frac{3}{4} x} {鈭�}_{0}^{2 - \frac{1}{2} x - \frac{2}{3} y} d z d y d x$

Evaluate the integral

$V = {鈭�}_{0}^{4} {鈭�}_{0}^{3 - \frac{3}{4} x} [{鈭�}_{0}^{2 - \frac{1}{2} x - \frac{2}{3} y} d z] d y d x V = {鈭�}_{0}^{4} [{鈭�}_{0}^{3 - \frac{3}{4} x} (2 - \frac{1}{2} x - \frac{2}{3} y) d y] d x V = {鈭�}_{0}^{4} {[2 y - \frac{1}{2} x y - \frac{2}{3} \frac{y^{2}}{2}]}_{0}^{3 - \frac{3}{4} x} V = {鈭�}_{0}^{4} [2 (3 - \frac{3}{4} x) - \frac{1}{2} x (3 - \frac{3}{4} x) - \frac{1}{3} {(3 - \frac{3}{4} x)}^{2}] V = {鈭�}_{0}^{4} (3 - \frac{3}{2} x + \frac{3}{16} x^{2}) d x V = {[3 x - \frac{3}{2} \frac{x^{2}}{2} + \frac{3}{16} \frac{x^{3}}{3}]}_{0}^{4} V = [12 - 12 + 4] V = 4 cubic units$

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91影视

Evaluating triple integrals: Each of the triple integrals that follows represents the volume of a solid. Sketch the solid and evaluate the integral.
${鈭�}_{0}^{4} {鈭�}_{0}^{3 - \frac{3}{4} x} {鈭�}_{0}^{2 - \frac{1}{2} x - \frac{2}{3} y} d z d y d x$

Short Answer

Step by step solution

Given

Evaluate the integral

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Evaluating triple integrals: Each of the triple integrals that follows represents the volume of a solid. Sketch the solid and evaluate the integral.鈭�04鈭�03-34x鈭�02-12x-23ydzdydx

Short Answer

Step by step solution

Given

Evaluate the integral

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Most popular questions from this chapter

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Logic and Functions

Pure Maths

Mechanics Maths

Discrete Mathematics

Geometry

Statistics

Study anywhere. Anytime. Across all devices.

Evaluating triple integrals: Each of the triple integrals that follows represents the volume of a solid. Sketch the solid and evaluate the integral.
${鈭�}_{0}^{4} {鈭�}_{0}^{3 - \frac{3}{4} x} {鈭�}_{0}^{2 - \frac{1}{2} x - \frac{2}{3} y} d z d y d x$